In the testing of test samples under tension and/or compression loads, it is necessary that the testing load applied in the testing machine is defined as precisely as possible. Thus, when the test sample is clamped in the testing machine it is necessary that bending loads are not applied to the test sample. Such bending loads can result, for example, when the test sample is canted or when the clamping heads are not precisely aligned with each other, for example, due to manufacturing tolerances at the clamping surfaces of the sample or due to similar influences, including thermal expansions and contractions. The application of the testing loads without any bending moments is especially important if the test sample is made of brittle materials such as ceramic materials which require an exactly central force application, since ceramic materials are especially sensitive to bending loads. In order to avoid the application of bending loads to such test samples it is necessary that the testing body is held in such a way in the clamping heads of the testing machine that the load application axis of the testing machine coincides with the longitudinal axis of the test sample. German Patent Publication (DE-OS) 3,731,460 discloses a clamping mechanism which is so adjustable that a test sample is clamped exactly in a centered manner. The known clamping mechanism comprises a clamping plate which holds the test sample and a clamping body which is connected to the loading device. For aligning the test sample, it is possible to adjust or align the clamping plate and the clamping body prior to the beginning of the test, by means of adjustment bolts which permit an adjustment relative to an axial displacement and an adjustment relative to an angular displacement or misalignment. However, during the loading of the test sample, it is possible that the initially aligned sample becomes misaligned relative to the loading axis due to possibly occurring micro-deformations of the test sample or of the clamping head. These length changes of the test sample that occur during the load application to the test sample cannot be compensated in the known apparatus because the clamping plate and the clamping head are adjusted relative to one another prior to the beginning of a testing sequence and because the once adjusted position is rigidly fixed by means of the adjustment bolts.